3. Let the random variable T denote the total time (in minutes) between a random

Question

3. Let the random variable T denote the total time (in minutes) between a randomly selected patient's arrival and departure from a health clinic. Let the random variable W deote the total time (in minutes) that a randomly selected patient spends in the clinic waiting room before seeing a health professional. The joint distribution of T and W is given by the following joint pdf. elsewhere where > 0 is an unknown constant. Note: for this problem, you may use (without derivation) any of the following results W~Exponential(9), T~Gamma(2,9), E(W) =, E(T) 29 (a) (3 points) Using the context of the scenario, explain in words why T and W must be dependent random variables.

Explanation / Answer

As given in the question, T is the total time interval between a patient's arrival and departure. W is the time spent in the waiting room. It is clear that the more time a patient spends in the waiting room, more will be the time interval between his or her arrival and departure. Therefore, W affects the value of T. Thus, T and W are dependent random variables.

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